On functional iteration methods for solving nonlinear matrix equations arising in queueing problems

The problem of the computation of the minimal nonnegative solution G of the nonlinear matrix equation X = [sum ]^+∞i=0Xⁱ Ai is considered. This problem arises in the numerical solution of M/G/1 type Markov chains, where Ai,i ⩾ 0, are nonnegative k x k matrices such that [sum ]^=∞i=0A is column stochastic. We analyse classical functional iteration methods, by estimating the rate of convergence, in relation to the spectral properties of the starting approximation matrix X0. Based on these new convergence results, we propose an effective method to choose a matrix X0, which drastically reduces the number of iterations; the additional cost needed to compute X0 is much less than the overall savings achieved by reducing the number of iterations.